We study a coevolving nonlinear voter model describing evolution of the states of the nodes and network topology simultaneously. An agent may switch its state by adopting the state of its neighbors by an interaction in which a number of neighbors influence an agent’s state in a nonlinear way. Nonlinearity of the interaction is measured by a parameter q. The network topology changes with a dynamics coupled to the dynamics of the state of the nodes by rewiring connections at a rate p. By analytical and numerical analysis we obtain a phase diagram in the q, p parameter space with three different phases: Dynamically active coexistence phase in a single component network, absorbing consensus phase in a single component network, and absorbing phase in a fragmented network. We find diverse transitions between these phases with different mechanisms: (i) continuous absorbing transition between active and fragmented phases and (ii) discontinuous transition between two absorbing phases, either in a single component network or a fragmented phase.